An EM waves basic diffraction question

[holess]

Could you please give me an explanation about this (it should be a basic question): why a slit with the width less than a EM wave wavelength does not allow the wave to go through?
I know Hygen's interference rule but somehow I don't see that if the slit is less than a wavelength it completely stops a wave. I found everybody cites that, but nowhere the explicit explanation (should be a special case of Hygen). Probably I didn't read the right books.
One application of that is: because the diameters of the little holes on microwave oven doors are less than their wavelength, it stops the microwaves and allows the interior to be seen.
Thank you veru much.

Ho

 

[turin]

You may be overgeneralizing. If you have a conducting material, then the material will reflect (to an extent) electromagnetic radiation. If this material is arranged in a mesh (like the microwave door), then the characteristic size of the mesh cell determines the characteristic cut-off frequency between reflections and transparencies. It is a matter of conservation. There is a certain amount of energy that gets reflected, the rest is transferred (or absorbed, but I'm assuming relatively high conductivity, which doesn't leave much room for absorption). If the cavities in the conducting material are large enough, then the material will not reflect so well, and therefore the wave passes through. If the cavities in the material are small, then the currents in the material can complete efficiently, and there is a high amount of reflection.

 

[holess]

hi.
thank you for your answer and your time. it sounds ok and reasonable
but i still have the problem to better understand it. i would appreciate if you could help me to make it clearer (for me).
in fact, everywhere it is explicitly stated that if the wavelength is less than a hole diameter it stops the wave. i would like to see where is that relation from. could you help me with that?
thank you very much.
ho

 

[turin]

Originally posted by holess
... if the wavelength is less than a hole diameter it stops the wave.

I'm not sure about that. I was thinking that you meant the reverse of this. Well, in this case, I don't know and can't be of any help at the moment. Did you mean "... if the wavelength is significantly greater than a hole diameter it stops the wave?"

 

[holess]

yes, i am sorry, you are right, i inverted the stuff:
"... if a diameter is less than a wavelength then it stops the wave .."
that was what i meant.
sorry about that and thank you very much.

 

[turin]

The idea is kind of like this (Electircal Engineer's version):
Consider a plane EM wave directly incident (angle of incidence = 0) on the conducting mesh. Think of one direction along this plane as the direction of the E-field. You can also think of this as the direction of a potential gradient. Since this is a wave, then in a particular spatially fixed plane, this gradient will "see-saw." Since the zero point of the electric potential is arbitrary (physically insignificant), then consider this to be in the center of one of the mesh holes. Therefore, this see-saw will "pivot" about the center of the mesh hole. Follow along the see-saw away from this pivot point until you get to the edge of the hole on either side. Now, the see-saw wants to force the conducting material of the mesh to be at different potentials, because of the see-saw action. The mesh disagrees with the wave, because it is a conductor, so it distributes its electron field to the appropriate places along the edge of the hole to maintain a constant potential. Well, if the hole is small compared to a wavelength, then the electron field has no problem keeping up with the see-saw potential of the wave, and essentially mimics it, but complementary. This is basically an image current with the same effect as one that would be induced in a continuous ground plane (and it is in fact induced for the same reason). So what you get is a reflection. However, if the holes are much greater than a wavelength, then the electron field has no chance, and the see-saw goes back and forth many times, faking the electron field out. The result is that the electron field just jiggles back and forth along the edge of the hole and the net effects at the center of the hole cancel out.

 

[holess]

thank you again very much.
i am going to think about that. actually, i like very much that kind of understanding. i have to think a little bit more to align my thoughts with your way of explaining.
and you reminded me about something that confuses me also: the angle of incidence. as i understood in your explanation E field strikes a hole in the same plane, kind of aside. what if a wave strikes with 90 degrees a plane that contains holes?
i really need that picture.
thank you very much for your time.

 

[turin]

Originally posted by holess
as i understood in your explanation E field strikes a hole in the same plane, kind of aside. what if a wave strikes with 90 degrees a plane that contains holes?

Can you try to rephrase this? I am not understanding what you are trying to ask.

 

[holess]

ok, i am sorry i was not clear. i meant,
if there is a hole and a wave propagates in the direction normal to it, like this (looked from a side):

-> |
-> |
wave -> hole
-> |
-> |

maybe that is not possible, or that doesn't change anything (what is the direction of microwaves in an oven, isn't it from "everywhere" to the hole)?
i definitely miss something to get over this.
thank you

 

[turin]

Originally posted by holess
i meant,
if there is a hole and a wave propagates in the direction normal to it, like this (looked from a side):

-> |
-> |
wave -> hole
-> |
-> |

That is the situation that I tried to explain. I suppose I was the one who was unclear. Sorry.

So anyway, now we need to identify what made you think otherwise. My "see-saws" were probably a little vague. I'll let you tell me what made you think that I was talking about something other than direct incidence (θ_incidence = 0^o).

 

[holess]

hi. thank you.
it was my fault. somehow i wandered off the definition of an angle of incidence (the angle that a direction line makes with a line perpendicular to a surface). ok, thank you again.
so, let me ask you this (i am very glad i have someone to ask).
on that way, when a hole is less than a wavelength, the effect is like we did not have a hole at all? the holes in a microwave oven are there just to let the light to go through because their sizes are greater then the light's wavelength.
what if we did not have a conducting material mesh, for example a wood mesh with the same characteristics? the waves would go through no matter how big is a wavelength?
and again, i believe there must be a connection to a Hygen's principle. i believe you know it. it talks about the same situation.
an EM wave directly incident to a hole and different wavelengths and diameters. what you explained to me must fit somewhere there. i would be so happy if you could put this situation in it. i couldn't find a-hole-less-than-a-wavelength situation nowhere.
thank you very much (maybe i ask too much).

 

[turin]

Originally posted by holess
... when a hole is less than a wavelength, the effect is like we did not have a hole at all? the holes in a microwave oven are there just to let the light to go through because their sizes are greater then the light's wavelength.

Well, I would be a bit careful about saying the effect is like that of a continuous conducting sheet. For example, there is an aggregate quantity called the skin depth. I'm pretty sure that the skin depth would be greater (at a given wavelength) if there are holes present (but this is just a guess). My reasoning is that the two situations are certainly different. Some effects are the same, but but there are probably other effects are different.

Originally posted by holess
what if we did not have a conducting material mesh, for example a wood mesh with the same characteristics? the waves would go through no matter how big is a wavelength?
and again, i believe there must be a connection to a Hygen's principle.
...
it talks about the same situation.
an EM wave directly incident to a hole and different wavelengths and diameters. what you explained to me must fit somewhere there.
...
i couldn't find a-hole-less-than-a-wavelength situation nowhere.

For one, the quasi-technical explanation that I gave for the conducting mesh does not apply here. I don't think that the wave would be stopped in this case. But it would be diffracted. The diffraction would be like a thin slit diffraction. In the case of a hole, it would be a 2-D Fourier transform as opposed to a 1-D transform.

If the medium is highly ponderable/polarizable, then maybe you would get the same effect/have the same explanation.

 

[holess]

thank you very much.
i know what is a "skin depth". actually i was reading something about the influence the em waves exhibit on humans and i tried to connect several things i read about.
could you recommend me a book or something else that explains things like you do about the em waves (or in general). i read several of them and i would like to have the additonal understanding.
i will try to find out when to apply hygen's principle and when not.
actually, no one from my books has ever mentioned a barrier (slit) material while explaining hygen.

thank you again very much.

 

[turin]

Originally posted by holess
could you recommend me a book or something else that explains things ... about the em waves ...

I wish. I have never come across an E&M book that I liked. I hear good things about Griffith's text, but I haven't read all the way through it, and what I have read didn't impress me all that much. Maybe someone else who has been listening could chime in and let us know. I would sure enjoy a good E&M text to use. DO NOT GO TO JACKSON'S BOOK! IT ABSOLUTELY SUCKS! (for anything but an antiquated pseudo-mathematical treatment)

Originally posted by holess
i will try to find out when to apply hygen's principle and when not.
actually, no one from my books has ever mentioned a barrier (slit) material while explaining hygen.

I know of Hyudgen's principle, but I don't really understand it. If I take the mechanism for granted, then it makes sense, but, do you have any light to shed on why the phase fronts act like isotropic point sources?